Mandelbrot cascade measures (w.r.t. uniform measures) were introduced by Mandelbrot [1]} to simulate the energy dissipation phenomenon in fully developed turbulences, and were developed by Kahane and Peyrière [2]}. When the random weights have Gaussian law, these measures are considered as a simplified model of the Gaussian multiplicative chaos built by Kahane [3]} (motivated by a model introduce by Mandelbrot [4]}), which has been playing a very important role in statistical physics and Liouville quantum gravity, see for example [5]}, [6]}, [7]}.
